Question

Am nevoie de ajutor va rog !​

Answer 1

$a)~|2x-8|=|x+2|~~~~~~~~~~~~~~~~|x|=\left\{\begin{array}{ccc}~~x,~daca~x\geq 0\\-x,~daca~x < 0\end{array}\right\\\\\\|2x-8|=\left\{\begin{array}{ccc}~~~2x-8,~daca~~2x-8\geq 0~~deci~~2x\geq 8,~adica~~ x\geq 4~ \\-2x+8,~daca~~2x-8 < 0~~deci~~2x < 8,~adica~~x < 4\end{array}\right\\\\\\|x+2|=\left\{\begin{array}{ccc}~~x+2,~daca~~x+2\geq 0~~deci~~x\geq -2\\-x-2,~daca~~x+2 < 0~~deci~~x < -2\end{array}\right$

$Pentru~~x\in(-\infty,-2):~-2x+8=-x-2\Rightarrow -2x+x=-2-8\Rightarrow \\\\\Rightarrow -x=-10\Rightarrow x=10\notin(-\infty,-2)\\\\Pentru~~x\in[-2,4]:~-2x+8=x+2 \Rightarrow -2x-x=2-8\Rightarrow \\\\\Rightarrow -3x=-6\Rightarrow x=\dfrac{6}{3}\Rightarrow x=2\in[-2,4]\\\\Pentru~~x\in [4,+\infty):~2x-8=x+2\Rightarrow 2x-x=2+8\Rightarrow \\\\\Rightarrow x=10\in[4,+\infty)\\\\Solutiile~ecuatiei~sunt~x\in\{2;10\}$

$b)~|2-3x|=|-x+3|\\\\|2-3x|=\left\{\begin{array}{ccc}~~2-3x,~daca~~2-3x\geq 0~~deci~~-3x\geq -2,~adica~~x\leq \dfrac{2}{3} \\\\-2+3x,~daca~~2-3x < 0~~deci~~-3x < -2,~adica~~x > \dfrac{2}{3} \end{array}\right\\\\\\|-x+3|=\left\{\begin{array}{ccc}-x+3,~daca~~-x+3\geq 0~~deci~~-x\geq -3,~adica~~x\leq 3\\~~x-3,~daca~~-x+3 < 0~~deci~~-x < -3,~adica~~x > 3\end{array}\right$

$Pentru~~x\in\Bigg(-\infty,\dfrac{2}{3}\Bigg]:~-2+3x=x-3\Rightarrow 3x-x=-3+2\Rightarrow \\\\\\\Rightarrow 2x=-1\Rightarrow x=-\dfrac{1}{2} \in\Bigg(-\infty,\dfrac{2}{3}\Bigg]\\\\\\Pentru~~x\in\Bigg(\dfrac{2}{3} ,~3\Bigg]:~-2+3x=-x+3\Rightarrow 3x+x=3+2\Rightarrow \\\\\\\Rightarrow 4x=5\Rightarrow x=\dfrac{5}{4} \in \Bigg(\dfrac{2}{3} ,~3\Bigg]\\\\\\Pentru~~x\in(3,+\infty):~2-3x=-x+3\Rightarrow -3x+x=3-2\Rightarrow \\\\\Rightarrow -2x=1\Rightarrow x=-\dfrac{1}{2} \not\in(3,+\infty)$

$Solutiile~ecuatiei~sunt~x\in\Bigg\{-\dfrac{1}{2};\dfrac{5}{4}\Bigg\}$